Chapter 4, Problem 1RQ

To determine

The performance specification for first-order systems.

Answer to Problem 1RQ

The performance specification for first-order systems is time constant.

Explanation of Solution

Introduction:

First order system are those systems whose input-output equation is a first order differential equation.

The analysis of the system can be done in two ways:

• Time domain analysis.
• Control domain analysis.

By using both the methods the working of dynamic control system can be analysed. There can be 1^st order, 2^nd order, 3^rd order ……n^th order systems.

The first order system is given as,

  $R(s)\to \boxed{\frac{1}{Ts+1}}\to Y(s)$

Here the response of the first order system can be easily found. Owing to the first order system, we get first order differential equations.

Here the time constant of the first order system is given by

$t=RC$ for resistors-capacitors circuit and $t=\frac{L}{R}$ for inductor-resistor circuit. The response of the first order system is given by,

  $V(t)=V(\infty )+\left(V(0)-V(\infty )\right)e^{-t/\tau }$

Where, all initial conditions refer to the time $t=t_0$ with constant input.

Thus, the performance specification for first-order systems is time constant. It tells us how fast the system responds.

Conclusion:

The performance specification for first-order systems is time constant.